Composite Materials Research Progress

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 45
implemented in the theoretical laws. The above-listed perspectives of research will be
successively considered in further works.
7.3. Improving the Calculation Time While Ensuring the Most Reliable
Predictions
The present work underlines the sometime existing opportunity to replace purely numerical
mathematical solutions by analytical forms enabling to significantly reduce both the time
required for designing the software and the time necessary for achieving one simulation. It
was demonstrated in this paper that Eshelby-Kröner could be, at least partially, presented as
an analytical model, while it was used for predicting mechanical states. Nevertheless, the
estimation of the macroscopic properties (elastic stiffness, coefficients of thermal expansion
and coefficients of moisture expansion) through the homogenization relations deduced from
this very model do still involve an implicit iterative procedure. It was already shown in the
literature by Welzel and his co-authors, that under specific conditions, it was possible to build
a model, numerically equivalent to Eshelby-Kröner model, from the combination of two
(separately less successful) other models (Welzel, 2002 ; Welzel et al. 2003). The concept is
similar to the idea based on empirical comparisons, historically proposed by Neerfeld
(Neerfeld, 1942) and Hill (Hill, 1952) to average Reuss and Voigt rough hypotheses in order
to get a numerically acceptable theoretical solution. In the field of micro-mechanical
modelling of composite materials, a combination of the two possible localization procedures
considered for Mori-Tanaka model in the present work would enable to numerically
reproduce the homogenized properties obtained from Eshelby-Kröner model. Building an
effective model from the two main ways of writing Mori-Tanaka model would mainly enable
to obtain closed-form solutions for the elastic stiffness tensor, instead of having to
numerically solve the iterative procedure involved in Eshelby-Kröner self-consistent model.
Thus, a coupling of this numerically effective solution for predicting realistic hygrothermomechanical
macroscopic properties to the already proposed in this very article analytical
forms for the local mechanical states would yield to a faster but still extremely reliable
innovative scale-transition approach for studying composite materials. The analytical forms
required for achieving the effective Mori-Tanaka model should be derived and published in
the near future.
References
Agbossou, A., Pastor J. (1997). Thermal stresses and thermal expansion coefficients of nlayered
fiber-reinforced composites, Composites Science and Technology, 57: 249-260.
Asaro, R. J. and Barnett, D. M. (1975). The non-uniform transformation strain problem for an
anisotropic ellipsoidal inclusion, Journal of the Mechanics and Physics of Solids, 23:
77-83.
Baptiste, D. (1996). Evolution des contraintes dans les matériaux composites – Modélisation
micromécanique du comportement des composites à renforts discontinus, In: Analyse des
contraintes résiduelles par diffraction des rayons X et des neutrons, Alain Lodini and
Michel Perrin (Editors).